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SNEAK PEEK: Hand of Fire by Judith Starkston
Hand of Fire by Judith Starkston
The Trojan War threatens Troy's allies and the Greek supply raids spread. A young healing priestess, designated as future queen, must defend her city against both divine anger and invading Greeks. She finds strength in visions of a handsome warrior god. Will that be enough when the half-immortal Achilles attacks? Hand of Fire, a tale of resilience and hope, blends history and legend in the untold story of Achilles's famous captive, Briseis.
Antiope's breath rasped like a distant wave scouring a rocky shore. Too faint to sustain life. Briseis squeezed her mother's hand, then balanced her mother's limp hand on her own, shifting each finger until the two matched up. When had her fingers grown as long as her mother's? It didn't mean she was ready to take on her mother's work alone. She rubbed gently, but Antiope's hand remained slack. Briseis shifted closer to her mother on the bed and adjusted the fleeces cushioning her mother's shoulders from the leather straps pulled across the bed's wooden frame. No response. What should I do, Mama? Tell me how to save you. You've taught me to be a healer from birth, but I don't know this, the one thing I have to know. Tell me.
Briseis leaned over to kiss her mother's forehead. Her lips pressed Antiope's skin—cool as bone—and Briseis's red-gold hair brushed against her mother's ashen cheeks. She shuddered at the contrast. The fever was gone. The single worry line had smoothed. These signs, reassuring at other times, tightened the knot in Briseis's chest—intimations of withdrawal, not healing. Her mother had let go.
Months ago she'd discovered her mother's secret illness. She'd begged her to fight it, tried every cure she could discover, even though her mother refused to offer advice, saying there was no point. Nothing Briseis did slowed her mother's decline. Now, Antiope would die if Briseis didn't strengthen that ragged breath.
Why hadn't her father and two oldest brothers stayed home today? The king had summoned them, but still …. All had seemed as usual in the morning, her mother no worse than any other day among the many she'd been ill. This sudden downturn had taken Briseis by surprise. Wake up, Mama, you can't die now. I won't let you.
Briseis glanced at the youngest of her brothers, Iatros, seated on a stool at the far side of her mother's bed. He twisted the hem of his tunic first in one direction and then in the other. Normally, she would have pulled him close and told him not to worry—she couldn't today. His fear was well founded. She turned toward her old nurse, Eurome, waiting for something useful to do, her soft, round figure sagging against the doorframe. What else can I try? Briseis felt the familiar tightening at her temples. She willed away the threatening headache. Not now. I need to think.
The storm that had blown in so suddenly threw wind and rain against the closed shutters of the clerestory windows. So late in the winter no one had expected such bad weather. Would it prevent her father and brothers from returning home? The pounding against the shutters sounded like some beast trying to break in—no ordinary storm. She imagined the squall building up force in some distant, dangerous place like Greece, blowing across the Aegean Sea and flinging itself against Mount Ida's flanks. The frenzy outside mocked her mother's stillness. Inside, the flickering light came from clay oil lamps, causing the geometric patterns frescoed on the mud-brick walls to lunge and recoil.
"She's barely breathing. What do we do?" Iatros said, leaning in to be closer to his mother.
Briseis shook her head. Though already sixteen and a year older than she was, he had a small, tense body and round cheeks, never yet shaved with a scraper. His gentle brown eyes and dark curls, startling against his worry-blanched face, deepened Briseis's dread of carrying on without their mother. Antiope had always sheltered Iatros in a manner unlike her treatment of his brothers. How would he fare without that guidance?
Briseis rubbed her eyes and blinked back the pain pressing behind them. Her hair, escaped as usual from its bronze butterfly clasp, annoyed her and she pushed it out of the way. Can't you give me a sign, Mama? We always worked together. Not anymore. Her mother's silence crushed down on her shoulders.
"What should we do?" Iatros said again.
Briseis looked at her mother's slack face. "We give her strength."
"If she doesn't breathe, how can she be strong?" Iatros walked around the bed and stood beside his sister, staring down at the still figure.
What have I missed all these months? Any moment, that small breath will stop. I have to do something. Briseis hunted in her mind for a way to make her mother strong again. She sifted through the teachings she learned from Antiope, rites and cures handed to generations of women in her family directly from the healing goddess Kamrusepa. She and her mother both served as priestesses to the goddess, and though Briseis herself had not yet grown comfortable with the temple rites, her mother always expressed closeness to the goddess, especially in the temple. Until the last year, Antiope's life vibrated with Kamrusepa's presence. This illness was inexplicable.
Briseis noticed a twitch at her mother's temple, then nothing. She pushed off the bed, straightening to her full height, taller than most men.
"We will have to help her breathe." She turned to the doorway and saw Eurome's eyes brighten. For as long as Briseis could remember, her nursemaid Eurome had been part of her life—pampering, scolding, teasing, teaching—constantly chattering, but always reassuringly there.
"Eurome, we'll light the brazier. Mama hates foul smells, so I haven't used this cure before, but I don't care now. We're going to make a plaster of wax and spices to strengthen her breath. That's what she would do."
Eurome rushed downstairs for a coal from the hearth, agile despite her weight.
Briseis reached for her mother's satchel, full of healing materials, and laid out supplies on a table beside the bed. She measured mustard seed and sulfur into a mortar and ground them. She glanced at her brother, his narrow shoulders hunched forward.
Briseis passed Iatros the satchel. "Find the ball of beeswax and soften it on the brazier."
Eurome returned with a coal in a long-handled bronze cup and lit the kindling in the brazier. She put an empty pot on to warm and then rubbed one of Antiope's hands, the smoothness of the sick woman's hand contrasting with the wrinkles of age and work. "Take my strength, Lady Antiope, all you need."
Iatros worked the wax in the pot and then used the edge of his tunic to pull it off the fire. "It's ready."
Briseis nodded. "Mix while I pour this in. Careful. It will sting your eyes, but that's why it will awaken Mama and strengthen the intake of her breath."
"Shouldn't I do that?" asked Eurome.
Iatros shook his head. "It's my job."
When they'd blended the yellow powder into the soft wax, Briseis undid the braided tie at the neck of her mother's sleeping tunic and exposed a small area. She touched the frail, ashen skin and Briseis's breath caught—her strong, proud mother so diminished now. Despite the powerful fumes that made their eyes water as Briseis spread the plaster on her mother's chest, Antiope's breathing did not deepen. Her eyes, however, fluttered open and she crinkled her nose in disgust.
Her mouth moved. Briseis bent close to hear the sighed fragments. "No more, daughter…. The hardest lesson… is accepting death." Antiope's chest rose and fell softly as she caught her breath. "No more for me to do. Your training… as healing priestess… is complete."
Antiope paused. "My need… to escape this pain is greater than your need… family's need… for me. I served the goddess… Now… you, Briseis." She turned her head toward Iatros. "Son…" He took her hand in both of his and kissed it. "You and your sister help each other." The bones along Antiope's cheeks drew her skin to a fragile thinness, almost translucent.
"You've devoted yourself to Kamrusepa all your life," Briseis said. "Why won't she save you and take away the pain?" Briseis's fingers knotted tightly into the curly fleece covering the bed frame.
Antiope closed her eyes a moment. Then she opened them and looked at her daughter. Her throat rattled with the effort to draw in more air. "Even goddesses… must accept the death of mortals dear to them… but she is here with me."
Antiope raised her hand a little from the bed. "Your father? Brothers?"
"Father hasn't returned, nor Adamas and Bienor. I sent a messenger after them as you asked."
"Ah, well…" Her mother's eyes closed, and she seemed to sink further into the bed.
Briseis slipped out of the room and stood at the top of the stairs, wanting to hear the noise of her father and brothers coming through the gate. She'd sent one of the menservants to tell them to come right away. In good weather it only took part of an afternoon to get from the walled city of Lyrnessos back to the estate, but with this storm …. When they left that morning, the sky was clear enough and her mother hadn't shown any alarming signs. They had no reason not to go, especially when the king needed her father's advice because of the rumors of Greek raids. Reports had come in of attacks on the towns north of their powerful ally Troy. Her father was the king's chief military advisor and he'd raised her two oldest brothers as warriors, so they had gone with him to the Council the king had called. But if they didn't get home soon, it would be too late.
Hearing nothing, Briseis turned back to her mother's room. When she held her fingers to her mother's mouth, she could barely feel the movement of breath. The burn of rage Briseis had felt for months flared hotter. Her mother had concealed her illness until she couldn't hide her pain any longer. They should have fought it together from the beginning. Her mother always said illness treated early was illness cured. When Antiope insisted they could do nothing, Briseis hunted for months for a cure among the clay tablets in her mother's library. While her mother had written, "Thus says Antiope" at the bottom of some of them, the tablets from past healers also filled the library. Briseis decided her mother hadn't had the strength to search through the records as she would have for any other ill person. But of the various teas and poultices Briseis tried, along with the many rites, none had improved Antiope's health.
Briseis believed her mother had given in to this illness, accepted defeat from the beginning. Illness generally came from the gods as punishment for violations against the gods' laws. In case her mother had neglected a sacrifice or some similar affront—any more serious sin seemed unlikely—Briseis performed a snake divination at the temple to ask Kamrusepa directly how they had offended the gods. But the swimming snakes had given only a muddled answer as they touched the words inscribed in the great basin. The snakes failed to identify anything Briseis could correct. Even before she'd tried the divination it had seemed impossible to Briseis that her mother could have sinned so greatly that Kamrusepa sent the illness, but giving in to the disease felt like a sin to Briseis. Her mother had resigned herself to death too easily, and the gods abandoned her because she did not love life enough—their gift to all. She needed to be dragged back to life.
Briseis had an idea. "You two stay with Mama. I need some supplies."
She ran downstairs to the back storerooms, the sound of the storm growing muted as she went deeper into the house with its thick walls. Once inside the library, the comforting odor of clay soothed her. Her mother, Briseis thought, was a mixture of lavender and earthy clay. She pulled tablets from the wooden pigeonholes, scanning the words formed with a reed stylus that her brothers said looked like bird tracks. She found it, "The Breath of Life Incantation." It hadn't made sense to her when she'd been required to copy it for practice three years ago, but it did now. Her heart felt light. She committed the rite to memory and tucked the palm-sized tablet back in its place.
She hurried through the megaron hall, the main room of the house with its two-storied ceiling and circular hearth, out to the main courtyard and into the kitchen opposite the stables. The wind-driven rain splattered under the portico's shelter.
The cook, a middle-aged woman with a kinder heart than her boney, hard face indicated, looked up in surprise from sorting lentils when Briseis appeared at the door.
"For Mama, hurry. I need honey, mint and sweet wine."
The cook quickly gathered everything on a tray, and Briseis carried it back upstairs. From the carved wooden chest next to the floor-to-ceiling loom in her mother's sitting room, she grabbed a sachet of lavender and a clay incantation jar shaped like a fig.
Iatros and Eurome looked up when she entered the sleeping chamber. She set down the tray on the table and leaned in close over her mother. Antiope's lips were parted, her eyes closed, their lids withered like fallen leaves in winter. The space between breaths felt impossibly long.
Iatros crouched by the bed, biting his upper lip, eyes fixed on his sister.
Briseis shifted her mother's legs aside and sat down. She closed her eyes and waited while the fear she felt emptied out with each breath she exhaled. The power of the ritual's words filled her mind. She called to Kamrusepa, praying for her to give power to this rite.
She opened her eyes and placed both hands on her mother's chest, then her head.
"Antiope, wife of Glaukos, mother of Bienor, Adamas, Iatros, and Briseis, you have heard death whisper in your ear. You have mistaken that whisper for the nurturing breath that flows in and out of every human being. You have gone after death. Return now. Hear the breath of life."
Briseis poured wine and honey into the fig jar, breathed into it, and then added the lavender and mint, crushing the leaves to release their scent as she held the jar close to her mother.
"Antiope, do you smell the spring? The time of new growth and blossoms? Remember the spring. Remember your children. Remember the sweetness of life. Remember that you love life. Take a strong breath."
Silently Briseis added, Come back, Mama, I need you. Remember how much I love you. Antiope sighed and her eyelids fluttered for a moment. Iatros cried out.
Briseis's heart leapt like a deer. "Mama!"
Daughter and son clung to their mother's hands. They waited for Antiope to open her eyes and reassure them that she would live. They listened for the slow rattle to quicken. Instead it faded, caught once, tangled in a last wisp of life, then fell silent.
Tears ran down Briseis's face, hot against her skin. Gradually her wet cheeks grew cold.
Later—how much later Briseis wasn't sure—Eurome gently pulled Iatros and Briseis from their mother's side. The old woman gathered them into her arms and held them tight against her generous bosom. Iatros sobbed, but Briseis's tears had burned away in anger. She raged at her mother, Kamrusepa, and death itself. In spite of the storm, she needed to run outdoors into the fields and woods. Something vital would boil over inside her if she did not escape from this room with its stink of sulfur and the sight of her lifeless mother.
She broke free from her nurse, raced down the stairs and out into the courtyard. As soon as she left the protection of the portico, the rain and wind hit her full force. She ran across the courtyard toward the tall wooden gates that protected the house from raiders. The gatekeeper had thrown them open in the hope that Glaukos and his sons would soon return.
Briseis pushed against the rain, running fast along the path through the fields of her father's estate toward the creek. The storm surrounded her. The clouds lay so low they seemed to swallow the foothills of Mount Ida that fingered down into her father's land. Mist cloaked even the lowest terraces of orchards on the slopes, and the forests higher up had disappeared entirely. Soon her clothes clung to her in soggy, cold folds that she tugged out of her way. The wind pulled most of her coppery hair from its clasp, blowing it about her face and up like a wild crown. The icy spears of rain drove hard against her face, and she felt relief at focusing on her discomfort.
She wanted her mother. Antiope taught that the hardest lesson for a healer is accepting death, but Briseis had already lost patients. She had learned that lesson. Her mother's death wasn't a lesson. It was personal, a searing pain. She turned her palms and face into the hard-pounding rain.
The leather soles of her shoes slipped on the muddy path. She stumbled over a broken branch near the creek. Fallen limbs lay all about. She ignored the menace and headed for the place where a huge oak reached its branches over the creek. She had come here often as a small child with her nurse to look for watercress for the kitchen. Later she came alone on hot summer days to cool her feet, enjoying the sound of the creek and the tree's leafy shelter. Now she wanted to throw her arms around the trunk, an anchor against her loss.
As she peered through the rain, there, unmistakably, was the sharp bend in the creek, but instead of the tree she saw a strange emptiness. The gray sky glowered at her where once the oak's branches had offered shelter. The creek's bank looked like an angry god had ripped out a huge piece of it and flung down stone and mud in disgust. Upturned roots reached helplessly toward their old home. The tree sprawled across the creek, and the water boiled up behind it, creeping toward her as it overflowed its banks.
She retreated from the muddy surge. Her wet skirt tangled her legs, but she yanked it out of the way and climbed onto a boulder high on the creek's bank. Her tunic stuck to her back. Rain ran down her face. She shivered with cold, but she didn't want to leave the storm. Staring at the chaos around her, she felt a kindred soul in this violence. She wanted to break and tear until she rooted out the grief that filled her at the thought of her mother's lifeless body in the upstairs chamber.
She sat on the boulder above the churning creek and curled her arms around her knees into a tight ball, her head tucked. How would the days ahead feel without her mother's steady presence? She wanted her mother with her for one thing especially—her marriage to the king's son, Mynes. That day loomed close and filled her with dread. She feared Mynes. Please let my fear be mistaken.
Why hadn't Kamrusepa protected her mother? She wanted to demand an answer, but that might prove dangerous. She must not anger the goddess. As the city's only healing priestess—no longer in training, whatever she might wish—Briseis now held the well-being of Lyrnessos in her hands: the fertility of crops and women, their health. She would pray and sacrifice to Kamrusepa at the Spring Festival, and the goddess would either listen or not. She must do all she could to be beloved of the goddess the way her mother had been.
Her mother's whisper returned. "Even goddesses must accept the death of mortals dear to them." Briseis unfolded herself and stood. She gathered her sodden skirt and leapt from the boulder, throwing herself into the wind, feeling the strength of it course through her body. That strength could dash her against the stones and yet she felt it inside like blood flowing through her, sustaining her. Did her goddess understand her grief with equal pain and offer this strength? She ran with the wind's force along the edge of the swollen creek and saw the image of tears pouring down Kamrusepa's cheeks. Then from the stores of tales she had loved throughout her childhood came other grieving goddesses, bent in sorrow, flooding her imagination, providing her with strange comfort that they suffered as she did and yet endured.
Amongst these images, one goddess stood out. Thetis crouched over the body of her warrior son, clawing at her cheeks while she howled. From the bard's tale, Briseis recognized Achilles in the bloodied figure, the greatest of the Greeks, with his red-gold hair, piercing green eyes, and fluidity born of the sea—except this hero still lived. Why this image of his death? Then she remembered that his divine mother had tried to burn away her infant son's mortality, to spare herself this eternal grief, each night laying him in the magic embers and searing away a little more frailty, creating the most invincible warrior of all. Her husband Peleus, a mere man, saw her rite and misunderstood the flames. He snatched the baby from the fire and through his mistake doomed him to die. His love-inspired ignorance would also doom Thetis to suffer eternal sorrow once that dread moment Briseis had imagined finally came. Even a goddess could not escape grief, no matter how hard she tried. Forever this searing pain, like Briseis's. Perhaps, Briseis allowed, Thetis's pain would be worse—a child lost, unbearable, unnatural. Perhaps the gods should not love any mortal too dearly.
Briseis turned toward home, letting the rain, cold and hard, obliterate her tears. She stumbled back to the house, her limbs spent, slow going amidst the maelstrom's p
Judith Starkston writes historical fiction and mysteries set in Troy and the Hittite Empire. Her novel, Hand of Fire, tells Briseis's story, the captive woman who sparked the bitter conflict between Achilles and Agamemnon in the Iliad. There was more to her than the handful of lines Homer gave her.
Starkston's upcoming mystery series features the indomitable Queen Puduhepa of the Hittites. On the first surviving peace treaty in history, Puduhepa pressed her seal beside her foe's, Rameses II, but she didn't realize until now that she was a sleuth.
Starkston is a classicist (B.A. University of California, Santa Cruz, M.A. Cornell University) who taught high school English, Latin and humanities. She and her husband have two grown children and live in Arizona with their golden retriever Socrates.
You can follow Judith Starkston on Twitter or Facebook or Google+. Starkston reviews historical fiction for the Historical Novels Review, the New York Journal of Books, The Poisoned Pen Blog, and on this website. She is a proud member of Desert Sleuths Chapter of Sisters in Crime and the Historical Novel Society. She leads the Arizona Chapter of HNS in order to foster an active historical fiction community in Phoenix.
My website URL http://www.judithstarkston.com/
Facebook: https://www.facebook.com/judy.starkston
Twitter: https://twitter.com/JudithStarkston
Goodreads: https://www.goodreads.com/author/show/6473981.Judith_Starkston
Amazon: http://www.amazon.com/Judith-Starkston/e/B008TSDBME
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The lingering effects of Hurricane Irma continued to negatively impact overall South East Florida Home Sales in October.
October was essentially a "catch-up month" for deals that were delayed the month before due to the storm.
In Palm Beach County, single-family home sales decreased 3.8 percent to 1,261 while the median sales price was up 4.8 percent to $325,000.
Active listings declined 3.1 percent and the supply of inventory dropped to 4.8 months. That's a seller's market.
A limited supply of affordable homes remains an issue in Palm Beach County. Of the 6,948 active listings, only 1,490 were priced below $300,000.
For the condo/townhouse market in Palm Beach, sales plummeted 8.5 percent to 901 while the median sales price surged 10.6 percent to $169,000 - Cash deals dropped 15.3 percent.
Active listings increased 3.2 percent and the supply of inventory increased to 5.6 months. Again….That's still a seller's market.


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Honor 20 Pro Global Launch Scheduled for August 2
Honor has announced the global release date of its Honor 20 Pro smartphone which is scheduled on August 2nd. The handset was launched in China and now it will be followed by the countries like Russia and then it will move to UK, France, Italy, Spain, Germany, Netherlands, Czech Republic, Finland, Poland, Malaysia, Egypt, Saudi Arabia, UAE, and few other markets.
The Honor 20 Pro was introduced along with the Honor 20 and the Pro moniker comes with sightly improved quad-camera setup along with a bigger battery. The device will be available for purchase in Russia for around 35,000 Rubel and the pricing for other countries will be known on the launch day. Both the smartphones will be coming out of the box with the latest Android 9 Pie operating system with the companies own UI on top.
Talking about the Honor 20 Pro, it will be coming with a 6.26-inch display with All-View Full HD+ display with a punch-hole placed on the top left corner that will be housing the front camera. Under the hood, there will be a HiSilicon Kirin 980 octa-core processor built on 7nm FinFET node coupled with 8GB of LPDDR4X RAM and 256GB of UFS2.1 flash storage. Furthermore, it will be coming with GPU Turbo 3.0 which will be improving the gaming experience.
For the cameras, there is a 48MP+12MP+8MP+2MP camera sensor setup on the back with Sony IMX586 sensor and with the second one being a super-wide-angle sensor with f/2.2 aperture, a telephoto lens, and a macro sensor. For selfies, the punch-hole display houses the 32MP camera with f/2.0 aperture size and the rear camera supports 30x digital zoom along with 3x lossless zoom.
The device will be coming with the support for Virtual Surround system which will be supporting most of the wired and wireless earphones along with Ultra-High ISO Sensitivity that should be offering a better low-light photography experience. There is a 40000mAh battery on the back with the support for 22.5W SuperCharge fast charging technology which will charge the phone form 0 to 50 percent in just 30 minutes. Comment in the section below if you have more queries on the same and stay tuned to PhoneRadar for more.
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redpajama_set_name: RedPajamaCommonCrawl


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LFA 60's Julius Anglickas: Put the Cepelinai Down, It's Time to Fight
UFC 224: Amanda Nunes and Raquel Pennington Set to Make History
Dane McGuire May 9, 2018
ONE Championship: Masters of Fate Preview and Predictions
Bryan Henderson November 4, 2019
Golden Boy on ESPN 2 Results: Hovhannisyan Overwhelms Rios, Rocha Scores First-Round Knockout
Zach Aittama March 9, 2018
Bellator 198: Fedor vs. Mir Weigh-in Video and Results
Mike Pendleton April 27, 2018
Amanda Nunes (Dave Mandel/Sherdog)
Riley Kontek Staff Writer
With the growing popularity of women's MMA, it is important to recognize these women with a rankings system similar to the men. Between the UFC's inclusion of a bantamweight division, its recent addition of a strawweight division and the all-female promotion of Invicta FC, more and more women are being exposed to casual and hardcore fans alike. Every month, Combat Press compiles the staff's individual rankings from featherweight to atomweight to create the Combat Press Women's MMA Rankings.
Featherweight Division (145 pounds)
Cristiane "Cyborg" Justino (1)
Marloes Coenen (2)
Julia Budd (3)
Charmaine Tweet (4)
Megan Anderson (5)
Annalisa Bucci (6)
Arlene Blencowe (7)
Latoya Walker (8)
Roberta Paim (9)
Daria Ibragimova (10)
It was a quiet end to 2016 for the featherweights in terms of fights, as none of the top-10 fighters were in action. However, there was a big announcement as the UFC will be adding the 145-pound division in the coming year. An inaugural title fight will take place at UFC 208. Unfortunately, Cristiane Justino will not be competing for that belt due to health concerns and a suspicious drug test. Instead, former bantamweight champion Holly Holm and Germaine de Randamie will be moving up a weight class to battle for the belt. Also in January, two top-five featherweights collide in the main event of Invicta FC 21, as Megan Anderson looks to pick up where 2016 left off against seasoned veteran Charmaine Tweet. Lastly, Arlene Blencowe drops to 135 pounds for a bout in her native country to take on invading Brazilian Amanda Lemos.
Bantamweight Division (135 pounds)
Amanda Nunes (1)
Valentina Shevchenko (2)
Holly Holm (3)
Ronda Rousey (4)
Julianna Pena (5)
Raquel Pennington (6)
Cat Zingano (7)
Miesha Tate (8)
Sara McMann (9)
Liz Carmouche (10)
The year of 2016 ended in crazy fashion for the bantamweight division, to say the least. In the main event of UFC 207, Ronda Rousey was unable to recapture the UFC bantamweight championship against Amanda Nunes. Instead, Nunes finished her in less than a minute. Also in December, Sara McMann defeated a returning Alexis Davis, who fought for the first time since taking time away from pregnancy. Looking into January, a Fox event will be headlined by the bantamweights, as Valentina Shevchenko and Julianna Pena look to punch their ticket to a title fight with the aforementioned Nunes.
Flyweight Division (125 pounds)
Jennifer Maia (1)
Vanessa Porto (2)
Roxanne Modafferi (3)
Julia Berezikova (4)
Sarah D'Alelio (6)
Rin Nakai (9)
DeAnna Bennett (5)
Agnieszka Niedzwiedz (8)
Kalindra Faria (7)
Poliana Botelho/Andrea Lee (10)
The flyweight division ended 2016 with major shakeups, as it continues to prosper as the most underrated division in women's MMA. Rin Nakai made a successful debut with the Rizin organization when she was able to beat Kanako Murata. Nakai's drop to 125 has been proven to be the correct move, as she is now 3-0 with three stoppages since her UFC release. Also, Agnieszka Niedzwiedz proved that her move to 125 was a fantastic idea, as she decimated Brazilian prospect Samara Santos to a brutal finish. Niedzwiedz is undefeated and one of the best prospects in the flyweight division. Looking into the first month of 2017, Andrea Lee will look to move up the ladder at Invicta FC 21 when she takes on veteran Jenny Liou. Also, DeAnna Bennett will take another crack at strawweight at that same event when she takes on Jodie Esquibel.
Strawweight Division (115 pounds)
Joanna Jędrzejczyk (1)
Claudia Gadelha (2)
Karolina Kowalkiewicz (3)
Carla Esparza (4)
Jessica Aguilar (5)
Jessica Andrade (6)
Rose Namajunas (7)
Michelle Waterson (10)
Alexa Grasso (8)
Joanne Calderwood (9)
The strawweight division is going strong going into 2017, as it has proven to be the most competitive and most exciting division in women's MMA. The only fight of the month was an important one, as Michelle Waterson and Paige VanZant headlined a Fox card. Waterson choked out the young star, cementing herself as a contender in a division full of sharks. The first month of 2017 will be a quiet one for the division, as none of the top-10 are in action.
Atomweight Division (105 pounds)
Ayaka Hamasaki (1)
Jinh Yu Frey (2)
Naho Sugiyama (3)
Herica Tiburcio (4)
Janaisa Morandin (5)
Julia Jones (6)
Tessa Simpson (7)
Satoko Shinashi (8)
Simona Soukupova (9)
Kyra Batara (NR)
Dropped out of the rankings: Kanna Asakura (10)
The atomweights finished up with a minor shake up in the month December, as two ranked fighters had bouts, but one was unable to stay in the rankings. On the positive side, Simona Soukupova made a quick turnaround from her Invicta loss to Herica Tiburcio to beat Anna Bezhenar in Europe. It kept her a top-10 fighter in a division that has plenty of up-and-coming talent. Kanna Asakura was unable to maintain her top-10 spot, as she was upset by Alyssa Garcia on the Rizin card in Japan. Currently, none of the top-10 fighters are scheduled to fight in January.
Editor's Note: Fighters are eligible to be ranked if they have competed in the last 18 months. Any fighter that chooses to switch weight classes will be ranked in their previous weight class until they have competed twice in their new division.
Amanda NunesBellator MMADeep JewelsFighter RankingsInvicta Fighting ChampionshipsMichelle WatersonPaige VanZantRonda RouseyUFCWomen's MMA
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Justyn Likes January 7, 2021


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Ioan Gherghel (18 augustus 1978, Baia Mare, Maramureș) is een internationale vlinderslagzwemmer uit Roemenië. Gherghel vertegenwoordigde zijn vaderland op de Olympische Zomerspelen 2000 in Sydney, op de Olympische Zomerspelen 2004 in Athene en op de Olympische Zomerspelen 2008 in Peking. Op de Europese kampioenschappen zwemmen 2000 in Helsinki behaalde hij zijn eerste medaille (brons).

Externe links

Roemeens zwemmer
Roemeens olympisch deelnemer

redpajama_set_name: RedPajamaWikipedia


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text:
\section{Introduction}
The study of geometrically frustrated  magnetic materials provides a fruitful path to the discovery of exotic physics.  In such materials, competing interactions between magnetic moments due to their localization on a highly symmetric lattice structure lead to large numbers of quasi-degenerate states.  Interesting physics resulting from such near degeneracies on frustrated lattices is thought to include: heavy fermion behavior in the spinel LiV$_2$O$_4$\cite{urano}, evidence for the existence of magnetic monopoles in the spin ice pyrochlores Dy$_2$Ti$_2$O$_7$\cite{Bramwell} and Ho$_2$Ti$_2$O$_7$\cite{Fennell1}, non-Fermi liquid behaviour in the distorted kagome based FeCrAs\cite{Wu}, spin liquid behavior in hyperkagome Na$_4$Ir$_3$O$_8$\cite{okamoto} and triangular lattice based $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$\cite{shimuzu}, and unusual yet to be understood Kondo physics in the pyrochlore Pr$_2$Ir$_2$O$_7$\cite{nakatsuji}.

While in traditional magnetic materials a two-dimensional net might arise in a layered lattice structure, a growing body of work has focused on designing model systems to test models of strongly correlated electron systems.  The study of two-dimensional frustrated systems has been augmented by the creation of artificial spin ice systems\cite{Wang,Qi}, where people have shown that it is possible to lithographically etch ferromagnetic islands in patterns that can be easily modeled.  From a theoretical perspective, the dipolar interactions crucial to such spin ice physics converge quickly in two dimensions, so that it is feasible to sum the dipolar interactions in real space.  Recent work on triangular optical lattices\cite{struck}, magnetic colloids\cite{steinbach} and macroscopic spin ice\cite{melladomacro} have generated further interest in two dimensional frustrated spin models.

\begin{figure}[here]
\label{figure1}
\includegraphics[scale=0.55]{figure1}
\caption{(Color online)(a) Selective removal of 1/9th of the sites of the triangular net gives an new edge-shared net.  The further removal of 1/7th of the bonds (indicated by dashed lines labeled $J_2$) results in the sorrel net shown in (b).  There are 8 sites in the unit cell, six of which are 4-coordinated $\{\alpha,\beta,\epsilon,\zeta,\eta,\iota\}$, and 2 of which are six coordinated $\{\gamma,\delta\}$.  (c) A photograph of oxalis regnelli triangularis, commonly known as wood sorrel with lines drawn atop it to guide the eye. (d) a common pattern on a doumbek
\label{figure1} }
\end{figure}

The Ising model on the two dimensional triangular net has been studied since the 1950's\cite{Wannier} and been shown to possess a large spin degeneracy as evidenced by a residual ground state entropy per spin of $\frac{S}{N} = 0.323$\cite{Nagai}. The simplest frustrated depletion of this edge-shared bipartite net removes every fourth spin in a regular fashion, substituting a non-magnetic atom for every second atom along each of the basis vectors, to create the kagome net.  The kagome net possesses a larger ground state degeneracy for the Ising model than the triangular net due to the corner-sharing nature of its bonds, as shown by an enlarged residual entropy per spin of $\frac{S}{N}= 0.50183$\cite{Kano}.  After many years of experimental searches, the kagome net has been experimentally realized both in quantum spin systems\cite{hiroi,shores,colman,schluter,okamotohiroi,ono} and artificial spin ice systems\cite{Qi}.  The triangular kagome net is a $\frac{7}{16}$th depleted triangular net, removing every fourth spin along the basis vectors and their neighboring hexagons.  Once again, the antiferromagnetic Ising model features a large residual entropy per spin, $\frac{S}{N} = \frac{1}{9}\ln(72)\approx 0.475185$.\cite{loh}  The maple leaf net\cite{betts} is a $\frac{1}{7}$th depleted triangular net, removing a triangular lattice formed by advancing along one basis vector by two units and a second by one unit in a manner similar to a knight in chess.  The Ising model on this edge-sharing net, which has coordination number 5 in contrast to the triangular (6) and triangular kagome and kagome (4) coordinations, appears to have not yet been studied.  Experimental candidates for both the triangular kagome\cite{mekata} and maple leaf\cite{cave,hawthorne,fennell} nets have been found

In this paper we treat the Ising model on the sorrel net, a  1/9th site depleted and further 1/7th bond depleted triangular corner-shared triangle net as described below in Sect. \ref{section1}.  By means of Monte Carlo simulations, we find that the classical antiferromagnetic Ising model on this net has a finite residual entropy per spin in the thermodynamic limit of $\frac{S}{N}$ = 0.48185$\pm$0.00008, slightly greater than the Pauling entropy estimate for this net, indicating that the Ising model on this net is more highly frustrated than on the triangular and triangular kagome nets, even approaching that on the kagome net. We calculate the heat capacity, acceptance rate, magnetic susceptibility and static structure factor as a function of temperature to aid the experimental identification of candidate systems.  We investigate the stability of this model to the presence of magnetic interactions on the removed bonds of ferromagnetic or antiferromagnetic sign, and show that for small bond strengths the residual entropy in the spin system approaches that of the sorrel net to quite low temperatures prior to undergoing a secondary transition to a state still possessing a finite residual entropy in the $T\rightarrow 0$ limit.

\section{A new two-dimensional lattice}\label{section1}

In this paper we introduce a 1/9th depleted triangular net as shown in Fig.\ref{figure1}(a) of symmetry P6mm.  One sees that the substitution of non-magnetic atoms (which themselves form a triangular lattice\cite{henley}) for every third atom along each of the basis vectors of the triangular lattice leads to an edge-shared net with sites 6-fold and 5-fold coordinated. Amazingly, the symmetric selective removal of 1/7 of the bonds on this lattice leads to a new highly frustrated {\it{corner-shared}} equilateral triangle two-dimensional net as shown in Fig. \ref{figure1}(b).  This creates a net reminiscent of {\it{oxalis regnelli triangularis}} (commonly known as wood sorrel) leaves as pictured in Fig.\ref{figure1}(c), and for this reason we will henceforth refer to this kagome-inspired net as the sorrel net.  Fig. \ref{figure1} (d) shows a picture of a doumbek.  The corner-shared triangle structure pictured in Fig. \ref{figure1} (b) is often found on the side of these drums, which are traditionally covered by mother-of-pearl, using simple Islamic tiling patterns.

  On the sorrel net the magnetic sites are four and six coordinated, such that one has two types of sites: one connecting two corner-shared triangles, another connecting three corner-shared triangles.  To our knowledge this net has not yet been found in nature.  It is our hope that our identification of this structure as an interesting frustrated lattice will encourage experimental groups to investigate materials naturally forming in this pattern, as has been the case with recent experimental searches for the maple leaf and kagome lattices.  Indeed, one recent study\cite{keene} has managed to achieve a $\frac{1}{9}$th depleted triangular lattice in a magnetic system, although, as we discuss below in Sec. \ref{section5}, the antiferromagnetic order seen may have resulted from additional hexagonal layers in this complex material.

\begin{table}
\begin{tabular}{|l|l|l|l|}
\hline
 Site & Position & Neighboring Sites: $J_1$ & $J_2$ \\
\hline
$\alpha$ & (0,0,0) &$\beta , \delta , \gamma_{ - {\bf{A}}_2} , \eta_{ - {\bf{A}}_1}$ & $\zeta_{-{\bf{A}}_2}$\\ \hline
$\beta$  &  $( \frac{a}{6}, \frac{ \sqrt{3} a }{6}, 0 )$ & $\alpha , \gamma , \delta , \iota_{ - {\bf{A}}_1}$ & $\epsilon$ \\ \hline
$\gamma$ & $( \frac{a}{3} , \frac{ \sqrt{3} a }{3}, 0 )$ & $\beta , \epsilon , \zeta , \alpha_{ + {\bf{A}}_2} , \iota_{ - {\bf{A}}_1} , \eta_{ - {\bf{A}}_1 + {\bf{A}}_2}$ &  \\ \hline
$\delta$ & $( \frac{a}{3} , 0 , 0 )$ & $\alpha ,\beta , \epsilon , \eta , \zeta_{ - {\bf{A}}_2} , \iota_{ - {\bf{A}}_2}$ & \\ \hline
$\epsilon$ & $( \frac{a}{2} ,\frac{ \sqrt{3} a }{6} , 0 )$ & $\gamma , \delta , \zeta , \eta$ & $\beta$\\ \hline
$\zeta$ & $( \frac{2 a }{ 3 } , \frac{ \sqrt{3} a }{3} , 0 )$ & $\gamma , \epsilon , \iota , \delta_{ + {\bf{A}}_2}$ & $\alpha_{+{\bf{A}}_2}$\\ \hline
$\eta$ &  $( \frac{2 a }{ 3 } , 0 , 0 )$ & $\delta , \epsilon , \alpha_{ + {\bf{A}}_1} , \gamma_{ + {\bf{A}}_1 - {\bf{A}}_2}$ & $\iota_{-{\bf{A}}_2}$\\ \hline
$\iota$ & $( a , \frac{ \sqrt{3} a }{3} , 0 )$ & $\zeta , \beta_{ + {\bf{A}}_1} , \gamma_{ + {\bf{A}}_1} , \delta_{ + {\bf{A}}_2}$ & $\eta_{+{\bf{A}}_2}$\\
\hline
\end{tabular}
\caption{  Sites, positions and nearest neighbors for atoms in the unit cell of the sorrel net. When not in the first unit cell neighbors are connected by basis vectors ${\bf{A}}_1 = (a,0,0)$ and ${\bf{A}}_2 = ( \frac{a}{2} , \frac{ \sqrt{3} a }{2} , 0 )$. $J_1 > 0$ is an antiferromagnetic bond between corner shared neighbors while $J_2$ connects the remaining nearest neighbors and is treated in three limits:$\{ -J_1 \leq J_2 < 0, J_2 = 0, 0 < J_2 \leq J_1 \}$. The limit $J_2 = J_1$ corresponds to the antiferromagnetic Ising model on the $\frac{1}{9}^{\text{th}}$ depleted triangular net. }
\end{table}

\section{Model}

  We consider an antiferromagnetic Ising model on the sorrel lattice with the Hamiltonian,
 \begin{equation}
 H = J_1\sum_{\langle ij \rangle}\!\sigma^z_{i}\sigma^z_{j}+J_2\sum_{\langle ik\rangle}\!\sigma^z_{i}\sigma^z_{k} \text{,}\label{equation8}
 \end{equation}
 where $J_1>0$ is an antiferromagnetic interaction between nearest neighbors on the corner-shared sorrel lattice, and $J_2$ is an interaction between nearest neighbor sites that when coupled create an edge-shared net. In our simulations we allow $J_2$ to vary in strength from $-J_1$ to $J_1$.

 \section{Method}

 We use the metropolis Monte Carlo algorithm to simulate $8\times L\times L$ nets subject to periodic boundary conditions.  Lattice sizes have been varied from $L$ = 1 to 18 on the sorrel lattice with $J_2$ =0 to ensure that the thermodynamic limit has been reached.  At each temperature, prior to averaging, $2\times 10^5$ to $10^6$ Monte Carlo steps have been used to equlibrate the spin system, with each average then taken over a further 2001 to 20001 Monte Carlo steps, where each step corresponds to on average attempting an update on each of the spin sites once.  To simulate the annealing\cite{Kirkpatrick} of an experimental material from high temperatures we have started with the temperature equal to 6000 $J_1$, reducing the temperature by multiplying by a factor of 0.99 for successive measurements.

\begin{figure}
\includegraphics[scale=0.35]{figure2}
 \caption{(Color online)(a) Acceptance rate versus temperature for $J_1=1$, $J_2=0$.  Representative lines for $L=1,2,9$ and $L=15$ are shown.  In the inset, we see that the zero temperature $L$=1 acceptance rate is considerably larger than the acceptance rate for $L=2$ and above.   Exact ground state results for $L=1,2$ calculated as discussed in the text agree with the $T\rightarrow 0$ limit of our simulations as shown by the horizontal lines in the inset and X's in the main figure. (b) Acceptance rate for $J_2\ne0$ ($L=4$) with hybrid spin flip processes allowable as discussed in Sect. \ref{sectionhsf}.  For $J_2>0$, as $T\rightarrow$0, twenty-five percent of all attempted single spin flip processes are accepted, corresponding to a magnetically ordered ground state with completely free spins at the 6-coordinated sites.  For $J_2<0$ single spin flip processes freeze out at low temperatures, while triple spin flip and double spin flip processes continue to allow the system access to all ground states at low temperatures.  \label{figure2}
\end{figure}

\subsection{Acceptance rate}

\subsubsection{Single spin flip}

For the pure sorrel net ($J_2=0$) we have carried out single spin flip Monte Carlo simulations and find a non-negligible acceptance rate for all $L$ ($\approx 41.5\%$ for $L\ge 2$) even in the zero temperature limit as shown in Fig. \ref{figure2} (a).  For small system sizes ($L=1,2$), we can find exact results for the zero temperature acceptance rates (0.425 and 0.4148376 respectively).  These are shown in the inset to Fig. \ref{figure2} (a) to agree well with our simulations.  The $L=2$ acceptance rate appears to lie slightly lower than the acceptance rate for higher values of $L$, which appear to reach a constant value well before $L=15$.

When $J_2>0$ the single spin flip process continues to admit flips on the 6 coordinated sites to the lowest temperatures, approaching a $25\%$ acceptance rate.  As the degrees of freedom associated to the spins at the remaining sites freeze out (as one expects from the detailed nature of the ground state configurations shown below, see Fig. \ref{figure4}), the acceptance rate is not a monotonic function of temperature, dipping below 25\% prior to recovering as $T\rightarrow 0$.

When $J_2<0$, single spin flip simulations are insufficient to allow the system access to its degenerate ground state, and the acceptance rate dies quickly as $T\rightarrow 0$, indicating a lack of ergodicity as has been commonly found in the dipolar spin ice systems, where loop algorithms have commonly been introduced.\cite{loops}

\subsubsection{Hybrid spin flip} \label{sectionhsf}

To address the concern that single spin flip processes alone may not allow our spin system to properly equilibrate for $J_2\ne 0$, given that in the ground state the spins joined by the $J_2$ bonds want to align(anti-align) when $J_2<(>)0$, we have developed a hybrid spin flip algorithm.   The addition of these flipping processes is in the spirit of loop algorithms\cite{loops} which have been deemed necessary in other frustrated magnetic systems to restore ergodicity to the system at low temperatures. To implement this hybrid spin flip algorithm, at each update attempt we randomly choose a site in the lattice.  If this site belongs to a $J_2$ bond, we randomly choose to either flip the spin (single spin flip (ssf)), or to flip the pair of $J_2$-bonded spins (double spin flip (dsf)).  If the chosen site belongs to a 6 coordinated site, then we randomly choose to either flip the spin (ssf) or to flip the spin and one of its randomly choosen nearest $J_2$ bonded pairs of neighboring spins (triple spin flip (tsf)).  Since 3/4 of all sites in the lattice belong to a $J_2$ pair, dsf processes are attempted on average during 3/8, tsf processes 1/8, and ssf processes 1/2 of the update attempts.  With this hybrid code, one sees (in Fig. \ref{figure2} (b)) that the total acceptance rate remains finite to zero temperature for all $J_2\ne 0$.

\paragraph{$J_2 > 0$}

As was found using ssf for an antiferromagnetic bond between the $J_2$ bonded sites, at the lowest temperatures only 1/4 of the spins remain active, and 1/4 of the ssf attempts within the hybrid spin flip are accepted.  However, the presence of dsf and tsf processes does allow the temperature evolution of the ssf acceptance rate to become smooth and monotonic.  This indicates that while the dsf and tsf processes effectively turn off once the system has achieved a partially ordered ground state (as described below), their presence at higher temperatures allows the system to remain in equilibrium as this order occurs.

\paragraph{$J_2 < 0$}

At low temperatures with a ferromagnetic bond between the $J_2$ bonded sites, ssf processes are no longer accepted.  However tsf and dsf processes allow the system to access its degenerate ground states and move from one to another.

\paragraph{$J_2 = 0$}

To test the validity of this approach, we have run hybrid spin flip processes for $J_2 = 0$.  These give results which agree with those of the ssf code.

\begin{figure}
\includegraphics[scale=0.32]{figure3}
\caption{(Color online)  (a) Heat capacity vs. temperature for $J_2$ =0.  For $L\ge 2$ heat capacities are very similar. For $J_2\ne0$ ((b) and (c)) a sharp peak is seen below the $J_2 = 0$ peak.  For $J_2 > 0$, (b), this peak sharpens with increasing system size.  For $J_2 < 0$, (c), both peaks show remarkably little variation with $L$.
\label{figure3}}
\end{figure}

\section{Results}

\subsection{Heat capacity}

As shown in Fig. \ref{figure3} (a), the heat capacity
\begin{equation}
C_V(T) = \frac{< \! E^2 \! > - < \! E \! >^2}{T^2}\text{,}\label{equation1}
\end{equation}
for the pure sorrel net shows a broad peak indicating the entrance of the system to a degenerate ground state which features two spins up(down) and one spin down(up) on each of the corner-shared triangles.  Here $E$ is the energy of a particular spin configuration on the lattice and $T$ is the temperature.   The heat capacity exhibits a mild dependence on system size, with the $L=1$ peak being noticeably larger than subsequent $L$.  There is little variation in the heat capacity for $L=2$ up to $L=18$.  In Fig. \ref{figure3} (a) we show heat capacity versus temperature curves for $L = 2$ and $L = 12$ to illustrate that finite size effects are likely small.

Turning on the coupling $J_2$, we see a sharp peak in the heat capacity well below this broad feature for both signs of $J_2$. As $J_2$ increases, this sharp feature moves to higher temperatures, broadens, and becomes less distinct from the peak representing the crossover to a two up (down) and one down (up) spin state.  In both cases, this secondary peak eliminates much of the entropy associated to the pure sorrel net at temperatures which increase with the magnitude of $J_2$.  As seen in Fig. \ref{figure3} (b), when $J_2 > 0$, as $L$ increases the second peak in the heat capacity sharpens as one might expect from a phase transition to a long range ordered state.  In strong contrast to this behavior, when $J_2 < 0$ as seen in Fig. \ref{figure3} (c), the second peak remains essentially unchanged as $L$ increases, indicating a crossover to a short-range ordered state.
 \begin{figure}
 \includegraphics[scale=0.1]{figure4}
 \caption{ A phase diagram illustrating the robustness of the sorrel lattice entropy to variations in $J_2$.  For $J_2>0$ at low temperatures one sees completely unconstrained spins on the 6 coordinated sites as illustrated.  For $J_2<0$ below the transition, satisfying the ferromagnetic $J_2$ bonds leaves the system with considerable remaining degeneracy.\label{figure4}}
 \end{figure}

\begin{figure}
\includegraphics[scale=0.35]{figure5}
\caption{(Color online) Heat capacity per spin at $J_2\ne 0$ minus the $J_2 = 0$ heat capacity per spin versus scaled temperature for $L = 2$.  (a) For $J_2 > 0$ the peak seems to occur at the same scaled temperature, indicating a linear dependence of the transition as shown in Fig. \ref{figure4}.  (b) For $J_2 < 0$ the crossover occurs at a lower temperature, and for small $J_2$ maintains a linear $J_2$ dependence at the critical temperature.\label{figure5}}
\end{figure}

\subsection{Phase diagram}

In Fig. \ref{figure4} we sketch the phase diagram for the Ising model on the generalized sorrel net.  In this plot, the dashed line represents the crossover from local disorder to each $J_1$ triangle having a net energy $-J_1$.  That is, below this line each triangle of the pure sorrel lattice adopts a spin structure with two spins up (down) and one spin down (up).  From the heat capacity, the temperature associated to this crossover corresponds to the maximum seen in the broad first peak of $C_v$ versus $T$, which appears not to change much as $J_2$ is turned on.  As the temperature is further decreased, for $J_2 > 0$ the spin system undergoes a transition to a long range partially ordered state, as pictured in Fig. \ref{figure4}.  In this state the 6-coordinated spin sites are completely free Ising spins, while all other nearest neighbor spin pairs feature antiparallel spins.

For $J_2 < 0$, as the temperature decreases there is a (size independent) crossover to a short-range ordered state, where across each $J_2$ bond the spins are parallel.  Every pair of parallel spins along $J_2$ bonds is possible, with the 6-coordinated spin site adopting a unique spin configuration antiparallel to the dominant spin configuration of its 3 nearest neighbor pairs.  One sees that such states are connected by the above-described dsf and tsf updates, and not by ssf updates.

At $J_2 = 0$, the spin system remains disordered to zero temperature classically.  As $J_2$ turns on, the peak corresponding to the transition/crossover to a long range ordered/short range ordered state occurs at proportionally larger temperatures, as indicated by the straight lines in our sketched phase diagram.  If the high temperature crossover and its associated temperature dependent heat capacity is truly independent of $J_2$, then by subtracting out the $J_2 = 0$ peak one would be left with the second peak alone.  Further, if one then scaled the temperature of several of these curves by their value of $J_2$, their peaks would be expected to coincide in the case that the transition temperatures scale linearly with $J_2$.  Such plots (for $L=2$) are shown in Fig. \ref{figure5} (a) (for $J_2 >0$) and (b) (for $J_2 < 0$). One sees that the temperature corresponding to the peaks of these curves does show a remarkable coincidence in both cases.  In fact, only the peak of $J_2 = -J_1$ appears to lie marginally below this linear temperature dependence.

\subsection{Entropy}

  At very high temperature one expects there to be essentially no correlations between the Ising spins, or $2^N$ equally weighted spin configurations for an entropy per spin of $\ln(2)$.   As one lowers the temperature, the residual entropy in the system decreases.  In ordered materials below the ordering transition temperature the entropy quickly approaches zero.  In magnetically frustrated spin systems entropy can remain to very low temperatures.  As such it is interesting to investigate the behavior of the residual entropy in the system
 \begin{equation}
 S(T) = S(T=\infty) - \int_T^\infty \! \frac{C_V(T)}{T} \, dT\text{,}\label{equation2}
 \end{equation}
and the zero temperature limit $S(T=0)$ of this quantity.

 \begin{figure}
 \includegraphics[scale=0.35]{figure6}
 \caption{(Color online) Residual entropy (as $T \rightarrow 0$) versus system size ($L$) is roughly constant beyond $L = 2$.  Shown are data from two different equilibration times with the same initial seeds and the effect on the heat capacity of a single spurious low temperature point in the heat capacity.  Simulations agree well with exact results at $L = 2$ and $L = 1$, which is shown on a different scale.  (Inset) Residual entropy versus $\frac{1}{L}$ for $L\ge 4$ is used to extract a residual entropy $\frac{S}{N} = 0.48185\pm 0.00008$ in the thermodynamic limit.\label{figure6}}
 \end{figure}
\subsubsection{Pauling estimate}

When $J_2 = 0$, if one makes the assumption that spins beyond their nearest neighbor are uncorrelated, then each spin site has two degrees of freedom constrained by the fraction of the possible spin configurations on each triangle which belong to the ground state, which for a triangle-based lattice is $\frac{6}{8}$.  On the sorrel net each spin belongs to either two or three triangles, so that there are $N_t = \frac{3N}{4}$ triangles in total.  The Pauling degeneracy of the ground state  is then $ 2^N(\frac{6}{8})^{N_t}=2^N(\frac{6}{8})^{ \frac{3N}{4}}$, hence the Pauling estimate for the entropy per spin is $\frac{S}{N} =\ln(2(\frac{3}{4})^{\frac{3}{4}})=\frac{1}{4}\ln(\frac{27}{4})\approx 0.477386$.
 \begin{figure}
 \includegraphics[scale=0.35]{figure7}
 \caption{(Color online) As discussed in the text, we have found exact results for the ground state residual entropy for $J_2 \ne 0$.  These agree well with MC simulations for (a) $J_2 > 0$ and (b) $J_2 < 0$.\label{figure7}}
 \end{figure}
\subsubsection{Exact}
When the number of spins is small, it is numerically possible to count the number of ground states that satisfy the ground state conditions for $J_2=0$.  For any spin configuration in the ground state two of the spins on any given triangle must be up(down) while the remaining spin is down(up).  So doing, for L=1 we find the ground state to have 40 distinct members while for L=2 it has 4921350 distinct states.  This indicates that the exact result for the residual entropy of the pure sorrel net is $\frac{S}{N} = 0.46111$ for L=1 and $\frac{S}{N} = 0.481534$ for L=2, both numbers which fall within our error bars for the residual entropy as seen in Fig. \ref{figure6}.

Adding $J_2\neq 0$, the available ground states additionally must satisfy parallel(antiparallel) spins across the $J_2$ bond when $J_2<(>)0$. This means that each ground state is a subset of the sorrel lattice states.  For $L=1$ this reduces the number of degenerate ground states to just 8 independent of the sign of $J_2$.  For $L=2$ when $J_2>0$ only 512 of the ground states satisfy the condition, while for $J_2<0$ there are 4096 ground states.   We see that these numerically extracted exact values are the $L = 1$ and $L = 2$ limits of general exact formulae for the $J_2 \ne 0$ entropy.  For $J_2>0$ all the ground states feature a completely free spin on the 6 coordinated sites, and only one of two global spin configurations for the remaining spin sites, indicating that we have $2^{2L^2+1}$ degenerate ground states, and that the entropy per spin should therefore weakly depend on $L$: $\frac{S}{N} = \frac{1}{8L^2}\ln(2^{2L^2+1})=(\frac{1}{4}+\frac{1}{8L^2})\ln(2)$.   For $J_2<0$, the ground states feature parallel spins on each of the $J_2$ bonds, meaning that each 6-coordinated site has a definite orientation antiparallel to the dominant spin type of its three neighboring $J_2$ bonded pairs of spins.  However, every combination of $J_2$ bond parallel spins is possible, so there are $2^{3L^2}$ ground states, for a ground state entropy per spin of $\frac{S}{N} = \frac{3\ln(2)}{8}$.  Hence for all $J_2 \ne 0$ we have exact results for the residual entropy.

\begin{figure}
\includegraphics[scale=0.35]{figure8}
\caption{(Color online) The entropy remaining in the system per spin as a function of temperature for $L = 4$.  At high temperatures, all curves asymptotically approach $\ln(2)$.  For $J_2 = 0$, the residual entropy smoothly decreases until reaching a constant low temperature value $\frac{S}{N} = 0.48173\pm 0.00005$.  For small $J_2$, the entropy remaining follows the $J_2 = 0$ curve to quite low temperatures prior to sharply downturning to the exact $T = 0$ residual entropies shown in the text.\label{figure8}}
\end{figure}

\subsubsection{Monte Carlo}

As shown in Fig. \ref{figure6}, the residual entropy per spin calculated by Monte Carlo simulations\cite{note3} agrees well with the exact results at $L=1$ and $L=2$ for the pure sorrel net.  Fig. \ref{figure6} additionally presents data using both the ssf and hsf codes.  We see that simulations from both codes agree with the exact results and give equally valid results at larger $L$.  We further note that for $L=2$ and beyond, the residual entropy per spin for this case ($J_2 = 0$) appears to have reached a constant value (0.48185$\pm$0.00008)\cite{fits} as a function of system size, indicating that the thermodynamic limit has been reached well before our maximum system size of $L=18$, which corresponds to 2592 spins.

As expected from our exact results, much of this residual entropy is quenched at very low temperatures when $J_2\neq 0$.  Our simulations indicate that the residual entropy per spin for $J_2<0$ remains constant as a function of $L$. All calculated residual entropies for $J_2 < 0$ agree within error of the exact results, except for the case $J_2 = -J_1$ and $L=4$, where a marginally lower entropy is found.  The reason for this small discrepancy has not been discovered as yet.  To calculate the errors for these results we numerically integrated four (or more at small $L$) heat capacities divided by temperature to find entropies versus temperature. We then found the average entropy at each temperature and its standard deviation.  As such, our stated errors may be a little small as they do not explicitly account for errors in the numerical integration procedure.

 For $J_2>0$ the residual entropy per spin is consistent with the exact results with all cases lying within two standard deviations.\cite{except}

In Fig. \ref{figure8} we plot the entropy versus temperature for various values of $J_2$ at $L=4$.  We see that at high temperatures as $J_2$ is turned on from zero, all the curves follow the $J_2=0$ entropy.  As the temperature continues to decrease, the larger magnitude $J_2$ simulations begin to substantially deviate from the $J_2 = 0$ entropy in a manner consistent with the phase diagram as extracted from the heat capacity in Fig. \ref{figure4}.  When $|J_2|$ is small, this deviation does not occur until quite low temperatures, and when it does occur, the transition is quite sharp to a state with the ground state entropy. If a system can be made for which the Ising model on the sorrel lattice with $J_2 << J_1$ is a good approximation, one would expect to see a plateau in the residual entropy of the pure sorrel lattice over a certain temperature range, as shown in the inset to Fig. \ref{figure8}.

\begin{figure}
\includegraphics[scale=0.42]{figure9}
\caption{For $L = 6$, an evaluation of the static structure factor at the indicated temperatures in reciprocal space for $J_2 = 0$, contour increments of 0.1 for (a) and 0.2 for (b).  The squished hexagon at the center represents the first Brillouin zone, which has been repeated to tile the reciprocal space.  The center of each hexagon indicates the number of reciprocal lattice vector steps to the centre from each side, the top center being (0,$\frac{16\pi}{\sqrt{3}a}$).  One sees diffuse weight about $\vec{q} = (\frac{4\pi}{a},0)$ and the 5 equivalent $C_6$ rotations about the origin at all temperatures.\label{figure9}}
\end{figure}

 \subsection{Spin-spin correlations}

The static structure factor provides a measure of the strength of the spin-spin correlations and their momentum dependence.  Where large single crystals of a material become available, the structure factor (convolved with an atomic form factor) allows a direct comparison between neutron scattering measurements and theoretical predictions.  Materials which order magnetically show the development of sharp, resolution-limited magnetic Bragg peaks at low temperatures.  In contrast, many frustrated magnetic systems show evidence of short range order in the form of disperse magnetic peaks concentrated around certain areas of reciprocal space.  Powder neutron scattering measurements average the angular dependence of such graphs, showing weight at finite wavenumbers.

\begin{figure}
\includegraphics[scale=0.35]{figure10}
\caption{(Color online) The ($L = 6$) static structure factor for $J_2 = 0$ at $T = 0.251 J_1$ and 10.05 $J_1$ along selected high symmetry directions.  Note that at both temperatures the weight remains symmetric about the origin to about $q = \frac{6\pi}{\sqrt{3}a}$.  Along $(0,q)$ the structure factor then repeats (note that this distance is 1.5 reciprocal lattice vectors).  Diffuse maxima occur about the points $\vec{q} = (\pm \frac{4\pi}{a},0)$ which are considerably less distinct at higher temperatures.\label{figure10}}
\end{figure}

Within Monte Carlo, the static structure factor is defined as,
\begin{equation}
{\mathcal{S}}(\vec{q}, T) = \frac{1}{N} \displaystyle\sum_{i,j} \! < \! \vec{s_i}(T) \cdot \vec{s_j}(T) \! > e^{i \vec{q} \cdot (\vec{r_i} - \vec{r_j})}\text{,}\label{equation3}
\end{equation}
where $N$ is the number of sites in the lattice $N=8L^2$, the summation runs over all pairs of spins on the lattice, the braces indicate the average over several MCS (20001 here) of the dot product between spins at sites $i$ and $j$, and $\vec{r_i} - \vec{r_j}$ is the vector between spin locations at sites $i$ and $j$.  For the sorrel lattice we have a hexagonal lattice with an eight site basis and reciprocal lattice vectors $\vec{b_1} = \frac{2\pi}{a}\left(1,-\frac{1}{\sqrt{3}}\right)$ and $\vec{b_2} = \frac{4\pi}{\sqrt{3} a}(0,1)$, where $a$ is the direct space lattice constant.  For a finite $8\times L\times L$ lattice, we can evaluate the structure factor at $\vec{q} = \frac{I}{L}\vec{b_1} + \frac{J}{L}\vec{b_2}$, where $I$ and $J$ are integers.  Taking $I$ and $J$ to run independently from $\{-4 L, 4 L\}$ allows us to gain a picture of the magnetic correlations well beyond the first Brillouin zone (shown as the central squished hexagon in Fig. \ref{figure9}).

\begin{figure}
\includegraphics[scale=0.42]{figure11}
\caption{For $L=6$ an evaluation of the static structure factor at the indicated temperatures for $J_2=J_1$ in reciprocal space. Contour increments of 0.2 in (a)  precede the resolution-limited Bragg peaks of (b).  The center of each hexagon indicates the number of reciprocal lattice vector steps to any point.  It is four reciprocal lattice vectors to the center from each side, the top center being $(0,\frac{16\pi}{\sqrt{3}a})$.\label{figure11}}
\end{figure}

\subsubsection{$J_2$ = 0}
When $J_2 = 0$, the structure factor weight is lowest in the vicinity of $\vec{q} = 0$, and isotropic at fixed $|q|$ as seen in Fig. \ref{figure9} and Fig. \ref{figure10}.  A disperse maximum is seen along the $(q,0)$ direction at $q = \pm\frac{4\pi}{a}$ and the equivalent $q = \pm\frac{8\pi}{a}$ and symmetry equivalent ($C_6$ rotation) points.  This corresponds to a linear combination of reciprocal lattice vectors, for instance, 2$\vec{b_1}$ + $\vec{b_2}$.  As the temperature is lowered, the relative intensity of this peak increases but the weight remains diffusely distributed around this point.

\begin{figure}
\includegraphics[scale=0.35]{figure12}
\caption{\label{figure12} For $J_2 = J_1$ the temperature dependence of the peak of the structure factor at ${\vec{q}}=(\frac{4\pi}{a},0)$ and equivalent points.  By $T = 0.251 J_1$ it has almost reached the ideal ordered value as discussed in the text.}
\end{figure}

\subsubsection{$J_2 > 0$}
For $J_2\ne 0$ at high temperatures the structure factor appears quite similar to the $J_2 = 0$ case, as seen in Fig. \ref{figure11} (a) and Fig. \ref{figure13} (a).  However, for $J_2 > 0$, as the temperature is lowered one sees the rapid shifting of weight to the maximal direction--which becomes an ordering wavevector as shown in Fig. \ref{figure11} (b).  In Fig. \ref{figure12} we present the temperature dependence of the structure factor at this wavevector.  One sees that it saturates at a value of approximately 121.70 by the temperature $T = 0.25 J_1$.  In this partially ordered state we have argued that the spins at the 6-coordinated sites are completely free to fluctuate while the remaining spins form a magnetically ordered state.  If one simply assumes a spin structure with no moments on the 6-coordinate sites and antiferromagnetically ordered classical spins on the remaining sites, one finds a structure factor of 121.50.  It is interesting to note that this is not the full moment $\frac{3}{4}\cdot 288 = 216$ corresponding to an alignment along a particular axis of the spin structure, as one is used to from both ordered ferromagnets and antiferromagnets.  In the partially ordered antiferromagnetic state although it is a two-sublattice structure, one sublattice gains a weighting of $\frac{1}{2}$ on its moments due to the geometry of the lattice in its structure factor, lowering the total moment.  Adding back the self correlation of free (unit magnitude) spins gains an additional 0.25 contribution to this idealized structure factor, suggesting that by $T=0.25J_1$ the ordering state may have nearly reached the ideal partially ordered structure.

\subsubsection{$J_2 < 0$}

When $J_2 < 0$, the low temperature spin correlations (Fig. \ref{figure13} (b)) show once more (like when $J_2 = 0$) a broad feature about $\vec{q} = (\pm 4\pi, 0)$, $\vec{q} = (\pm 8\pi, 0)$ and 6-fold symmetry related points.  This feature sharpens as the temperature lowers, as can be seen in Fig. \ref{figure14}. Additionally, the contours appear less circular, more hexagonal than the $J_2 = 0$ case.

\begin{figure}
\includegraphics[scale=0.42]{figure13}
\caption{For $L=6$ an evaluation of the static structure factor at indicated temperatures for $J_2 = - J_1$ in reciprocal space, contour increments of 0.1 (a) and 0.2  (b). The center of each hexagon indicates the number of reciprocal lattice vector steps to any point.  It is four reciprocal lattice vectors to the center from each side, the top center being $(0,\frac{16\pi}{\sqrt{3}a})$.  (a) At $T = 5.07 J_1$, the highest weight lies at $(0,\pm \frac{6\pi}{\sqrt{3}a})$ and equivalent positions, but considerable weight lies at $\vec{q} = (\pm\frac{4\pi}{a},0)$ which grows at low $T$ to form diffuse peaks.\label{figure13}}
\end{figure}

\begin{figure}
\includegraphics[scale=0.35]{figure14}
\caption{The structure factor at low and high temperatures is now roughly symmetric only from $-\frac{4\pi}{\sqrt{3}a}$ to $\frac{4\pi}{\sqrt{3}a}$ and is less rounded than at $J_2 = 0$ (Fig. \ref{figure10}).  A diffuse maxima occurs at $\vec{q} = (\pm \frac{4\pi}{a},0)$ at low temperatures but washes out at higher temperatures. \label{figure14}}
\end{figure}

\subsection{Magnetic susceptibility}
The static magnetic susceptibility is one of the first measurements made on new magnetic materials.  It is calculated as,
\begin{equation}
\chi(T) = \frac{{\mathcal{S}}(\vec{q}=\vec{0}, T)}{T} = \frac{1}{N \, T} \displaystyle\sum_{i,j} \! < \! \vec{s_i}(T) \cdot \vec{s_j}(T) \! >\text{.}\label{equation4}
\end{equation}
Plots of the inverse magnetic susceptibility versus temperature are shown in Fig. \ref{figure15}.  At high temperatures (Fig. \ref{figure15} (a)), the inverse susceptibility shows a Curie-Weiss form with a large negative Curie-Weiss constant indicating the dominance of antiferromagnetic couplings between nearest neighboring sites.  A characteristic feature of frustrated magnets is that their magnetic susceptibility shows no signs of magnetic order to well below the Curie-Weiss temperature.  Unlike ordering antiferromagnets which exhibit non-monotonic susceptibilities at the N{\'e}el temperature, for $-J_1\le J_2\le J_1$, the inverse susceptibility monotonically decreases with the temperature.  It is not uncommon for frustrated materials to have defects or uncorrelated ``orphan spins'', such that the susceptibility diverges more quickly at low temperatures, as a Curie law.  In this system we see an intrinsic property of Ising spins on the sorrel lattice is a low temperature Curie tail.

\begin{figure}
\includegraphics[scale=0.30]{figure15}
\caption{(Color online) Inverse magnetic susceptibility versus temperature. (a) A high temperature Curie-Weiss intercept gets increasingly negative as $J_2$ increases.  Shown are $L =2$ and $L=4$ for all curves, and additionally $L=6$ for $J_2 = -J_1$.  There is very little system size-dependence of any part of the susceptibility beyond $L=2$.  (b) Variation of $J_2$ for $L=2$.  Curves asymptote at low temperatures towards different Curie laws depending on the sign of $J_2$, approaching less quickly the weaker $J_2$ is. (c) The Curie law for $J_2 = 0$ slightly differs from that for $J_2 > 0$.   \label{figure15}}
\end{figure}

\subsubsection{$J_2 = 0$}

When $J_2 = 0$, the inverse susceptibility crosses over from a high temperature Curie-Weiss law to a low temperature Curie law.  For $L=2$ we find $\chi^{-1} = (3.9134\pm 0.0007) T$.  This likely indicates that a substantial fraction of the spins at the 6-coordinated sites are freely flipping at low temperatures.

\subsubsection{$J_2 > 0$}

For $J_2 > 0$, this is certainly true.  At the lowest temperatures (see Fig. \ref{figure15}(b)) all $J_2 > 0$ curves that we have investigated approach the same Curie law.  For $J_2 = 1.0$, and $L = 4$ we find this to be $\chi^{-1} = (4.000\pm 0.001)T$.  This is consistent with $\frac{1}{4}$ of the spins being completely free Ising spins, while the remaining spins are ordered.  Interestingly, one does see that at temperatures of order $-\theta_{CW} = (5.97\pm0.02)J_1$, the inverse susceptibility begins to deviate upwards from its Curie-Weiss fit, as one might expect from an ordering antiferromagnet.  However, as it begins to order, it frees the 6-coordinated spins, causing $\chi^{-1}$ to eventually downturn at low temperatures once more.  As shown in Fig. \ref{figure15} (a), there is surprisingly little system size-dependence to the magnetic susceptibility, so we have chosen to show $L=2$ plots in Fig. \ref{figure15} (b).

\subsubsection{$J_2 < 0$}

The low temperature susceptibility for $J_2 < 0$ follows a distinctly different Curie law.  For $L = 2$ it is not hard to write out all possible ground state spin structures consistent with every $J_2$ bond having aligned spins.  So doing, one finds total spins ranging from $S = 0$ to $S=16$ and their degeneracies as shown in Table \ref{tabletwo}.  Writing the magnetization in the limiting case $B\rightarrow 0$ in terms of the magnetic moment per spin state and the partition function allows us to evaluate the magnetic susceptibility and find an exact result, $\chi^{-1}_{L=2} =\frac{32}{14}T\approx 2.2857 T$.  Our Monte Carlo results\cite{notechi} at $L =2 $ agree with this exact calculation for $J_2 = -J_1$ at low temperatures, $\chi^{-1}(T) = (2.28689\pm 0.00059)T - (7\pm 30)\times 10^{-5}$.  The low temperature asymptotes for $J<0$ for higher $L$ appear to agree with this result as well.

\begin{table}
\begin{tabular}{|l|l|}
\hline
Total spin & Degeneracy  \\
\hline
$\pm$16& 1\\
\hline
$\pm$12& 12\\
\hline
$\pm$10&24\\
\hline
$\pm$8&96\\
\hline
$\pm$6&200\\
\hline
$\pm$4&516\\
\hline
$\pm$2&672\\
\hline
0&1054\\
\hline

\end{tabular}
\caption{Degeneracies of the ground state spin configurations satisfying parallel spins on every pair of $J_2$ bonded spins as a function of the total spin of the $L = 2$ spin configuration.  \label{tabletwo} }
\end{table}

\section{Discussion}\label{section5}

\subsection{Non-magnetic depletion}

In this work we have introduced the sorrel net and provided the first study of frustrated magnetism on this depleted triangular lattice.  The regular substitution of non-magnetic atoms into a frustrated magnetic sublattice of a material is a useful way of predicting and perhaps generating new candidate exotic materials.  One such material is the quantum spin liquid candidate Na$_4$Ir$_3$O$_8$\cite{okamoto}, which features $s=\frac{1}{2}$ Ir atoms at three of the four corners of the corner-shared tetrahedral lattice (common to the pyrochlore and B-site spinel geometries).  The Ir atoms in this material are arranged to create a three-dimensional corner-shared triangle hyperkagome lattice.  Non-magnetic Na atoms sit at the fourth corner of each tetrahedron, and whether because of substantial size differences\cite{Shannon} between Na$^+$ and Ir$^{4+}$ ions or electrostatic interactions with surrounding atoms, the Na ions spread throughout the lattice in a regular pattern.

\subsection{Evidence for depleted triangular lattices}

In this context, it is interesting to note that triangular lattice depletions of the triangular lattice not only yield the ($\frac{1}{9}$th doped) sorrel lattice presented here, but also a number of frustrated lattices for which experimental candidates have already been found. An impressive collection of pictures of possible trigonally symmetric lattices derived by periodic site depletion is presented in the electronic supplementary information to Ref. \onlinecite{keene}. Most notably, the much studied kagome lattice, a $\frac{1}{4}$ doped triangular lattice, has recently been realized experimentally in the materials volborthite\cite{hiroi}, herbertsmithite\cite{shores}, kapellasite\cite{colman}, haydeeite\cite{schluter}, vesignieite\cite{okamotohiroi}, and Cs$_2$Cu$_3$SnF$_{12}$\cite{ono}.  Many of these minerals are Cu based as experimentalists search for low spin quantum spin liquid candidates.  Additionally, artificial spin ice on the kagome lattice was first produced by Qi {\it{et al.}}\cite{Qi}

 The $\frac{1}{7}$th depleted triangular lattice, the maple leaf lattice, was introduced in 1995 and has candidate materials: M$_x$[Fe(O$_2$CCH$_2$)$_2$NCH$_2$PO$_3$]$_6\cdot n$H$_2$0 where $x =11$ if M is Na or K, and $x =10$ if M is Rb\cite{cave} and spangolite\cite{hawthorne,fennell}.  Even the triangular kagome lattice realized by the Cu atoms\cite{mekata} in Cu$_9$X$_2$(2-carboxypentonic acid)$_6$$\cdot$xH$_2$0, where $X$ is F, Cl or Br, studied by Loh {\it{et al.}}\cite{loh} is a triangular lattice $\frac{7}{16}$th depletion of a triangular lattice, where each point of the triangular lattice removes 7 spins.

The natural extension of the doping of the triangular lattice from the kagome lattice (which removes every second spin along the reciprocal lattice vectors), to the sorrel lattice (which removes every third spin), would lead to the removal of every fourth spin along the reciprocal lattice vectors for a $\frac{1}{16}$th doping.  Interestingly, for such a lattice it is not possible to remove bonds to create a new corner-sharing triangle lattice.  However, the selective removal of edge-sharing bonds analogous to setting $J_2 = 0$ on the sorrel lattice, would create an interesting potentially frustrated lattice--featuring a star of David decoration inside each empty hexagon of the triangular kagome lattice.

\subsection{A $\frac{1}{9}$th depleted triangular lattice}

Is it possible to create a regular $\frac{1}{9}$th site depleted triangular magnetic lattice in an insulator?  We were pleasantly surprised to learn that this has indeed recently been achieved in a new cobalt hydroxide oxalate Co$_{12}$(OH)$_{18}$(C$_2$O$_4$)$_3$(C$_4$N$_2$H$_{10}$) by Keene {\it{et al.}}\cite{keene}.  In this material the planar Co atoms are octahedrally oxygen coordinated, with $\frac{1}{9}$ of the octahedra empty.  Unfortunately, above and below the empty (in-plane) octahedra are Co atoms.   Three oxygen atoms from the tetrahedra forming the base of a tetrahedral coordination of these Co atoms with nitrogen providing the apical atom of the tetrahedra. It is not immediately clear what effect couplings to such tetrahedral Co atoms might create, although even a weak coupling between the tetrahedral Co atoms might make a strong (ferromagnetic) perturbation on the hexagon of Co atoms nearest to them. Additionally, this layered material has a 2 dimensional honeycomb network of oxalate mediated couplings between Co ions.  Magnetically, this material shows interesting, although perhaps not frustrated behavior, apparently antiferromagnetically ordering at $T_c = 23.5\pm0.5 K$ despite an antiferromagnetic Weiss constant of only about 3.6 K. Given that the authors\cite{keene} expect an antiferromagnetic coupling in the honeycomb layer of 18$\pm 3$K from previous work\cite{andrefwithin}, it seems natural to ask whether or not the bonds of the sorrel layers in this material indeed have antiferromagnetic couplings, or are contributing to lowering the effective Weiss constant.

One expects that the predominant interaction between Co atoms in the plane is via superexchange through the joining oxygen sites, and that as the Co-O-Co bond varies from 180$^o$ to 90$^o$ at some angle (roughly 120$^o$ in the cuprates) ferromagnetic exchange will be favored over antiferromagnetic exchange.  Of the four in plane bonds presented by the authors, $\{{\mathcal{J}}_1,{\mathcal{J}}_2,{\mathcal{J}}_3,{\mathcal{J}}_4\}$ involving planar Co atoms, our J$_1$ corresponds to the assumption of equal bond strengths on their ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$, our $J_2$ corresponds to their ${\mathcal{J}}_3$ and their ${\mathcal{J}}_4$ is a new bond to the tetragonal Co atoms.  While it would be interesting to carry out a detailed calculation of these exchange integrals in future work, na{\"i}vely from bond angle considerations\cite{bondangles} one would say that ${\mathcal{J}}_4$  is likely to be weakly antiferromagnetic,  ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$  may well be weakly ferromagnetic, and ${\mathcal{J}}_3$  is likely to be weakly ferromagnetic. As such, one might expect that this system falls in a regime not covered in this work, of ferromagnetic $J_1$ and $J_2$.   It would certainly also be interesting if a local probe such as oxygen NMR could be used to determine the signs of the magnetic couplings within the plane.  If, on the other hand, it were shown that this material possesses an antiferromagnetic $J_1$, one might hope to exploit the differences between the atomic sizes of octahedrally and tetrahedrally coordinated Co$^{2+}$ ions to design a non-magnetic substitution.  For example, according to Shannon\cite{shannon1}, tetrahedrally coordinated Mg$^{2+}$ is very close in size to tetrahedrally coordinated Co$^{2+}$, while the octahedral coordinated ions of both species are of quite different sizes, suggesting that if it were possible to substitute Mg atoms for Co atoms, the Mg atoms might preferentially enter the tetrahedral site, removing some complicating non-frustrated magnetic physics.

\subsection{Designing for antiferromagnetic superexchange}

While we view the solution of the Ising model on the sorrel lattice as a simple test whether or not antiferromagnetism would be frustrated on this net, it is interesting to speculate on how this, or related models might be realized in an experimental system.  In particular, one might wonder how magnetic exchange processes could lead to an antiferromagnetic $J_1$ and a weak, non-existent, or ferromagnetic $J_2$, even if the magnetic atoms of a crystal could be arranged into the $\frac{1}{9}$th doped triangular net.  One could imagine\cite{possible} superexchange pathways via say oxygen atoms living at the midpoints of the $J_1$ bonds leading to a strong antiferromagnetic coupling on these bonds alone.  Indeed even oxygen atoms lying at the center of the $J_1$ triangles would likely lead to weakly antiferromagnetic $J_1$ bonds.  Such interactions might be expected to produce at least roughly equal strength bonds around both the 6-coordinated and 4-coordinated spin sites, as we have assumed in this work.  To additionally realize Ising spins would likely require strong crystal field anisotropies in the out of plane direction, leading to a magnetically easy axis.

\subsection{A two-dimensional structure}

That the sorrel net is two-dimensional opens a host of possible realizations for further study.  For example, recent advances in the study of spinless bosons trapped by optical lattices have allowed the simulation of models of frustrated magnetism on the triangular lattice with couplings of various magnitudes.\cite{struck}  Could such studies be extended to the doped triangular lattices?  From soft condensed matter, the study of magnetic colloids interacting via the dipolar interaction offers the possibility of real time manipulation of a lattice using optical tweezers to perhaps create a mesoscopic frustrated lattice of choice.{\cite{steinbach}}

While it may be currently unrealistic\cite{nogo} to expect the creation of a direct artificial realization of a dipolar Ising model on the sorrel lattice with moments pointing out of the plane,  the construction of a coplanar artificial spin ice structure should be feasible.  As for (lithographically etched) realizations of artificial spin ice on the kagome lattice\cite{}, a permalloy island could be to chosen to point locally along the axis of symmetry of the triangle.  While such islands could join the centers of the triangles about the 4-coordinated spins sites, a new and interesting feature for study would arise at the 6-coordinated spin sites.  Three overlapping ferromagnetic islands (with a $C_6$ symmetry) would mimic a multiorbital local  Ising spin at these vertices, producing a novel interplay between short range contact terms (Hund's coupling and ferromagnetic interactions) with the long range dipolar interaction.  We have begun \cite{ianandme} the study of ferromagnetic interactions between such local Ising spins in the presence of Hund's coupling terms and find, at least to this approximation, that degeneracy remains in the ground state of the sorrel lattice.  Unfortunately, this multiorbital nature appears to prevent the creation of a macroscopic spin ice along the lines of that created by Mellado {\it{et al.}}\cite{melladomacro}

The question of what quantum magnetism will do on the sorrel lattice is left for future work.   When considering frustrated quantum spin systems, perhaps the simplest model is that of the quantum transverse Ising model\cite{moessner}, the study of which is considerably simplified for low dimensional lattices.
  We\cite{jarrettandme} have begun to investigate classical dimer models on the sorrel lattice using the Pfaffian approach.  The hope is that as on the triangular lattice, the classical and quantum dimer models may be related, and there may be a mapping\cite{moessner} to the quantum transverse Ising model.

\section{Conclusions}

We have studied the antiferromagnetic Ising model on a new corner-shared equilateral triangle net, finding a large finite residual entropy $\frac{S}{N}$ =0.48185$\pm$0.00008, slightly larger than the corresponding Pauling entropy $\frac{S}{N}=\frac{1}{4}\ln(\frac{27}{4})\approx 0.477386$.  We have shown that the spin-spin correlations  remain disordered to the lowest temperature, with the structure factor showing broad disperse peaks about $\vec{q} = (\frac{4\pi}{a},0)$ and $C_6$ rotations.  The magnetic susceptibility indicates that strong antiferromagnetic correlations at high temperatures give way to short range ordering at low temperatures, with a Curie tail with slope close to that one would expect were $\frac{1}{4}$ of the Ising spins completely free.  While we have not proposed an exact solution, our results are remarkably independent of system size, indicating that the thermodynamic limit is quickly reached.

To this corner-shared triangle sorrel net we have added edge-sharing bonds ($J_2$) which are likely to remain present in physical realizations of the doped triangular lattice, and investigated the phase diagram produced by varying the magnitude and sign of this coupling constant.

For antiferromagnetic edge-sharing couplings ($J_2 > 0$), we have shown that at low temperatures our spin system adopts a partially ordered state with Ising spins at the 6-coordinated sites completely free, and all other neighboring spins antiferromagnetically correlated, resulting in sharp magnetic Bragg peaks at $\vec{q} = (\frac{4\pi}{a}, 0)$ and $C_6$ rotations.  This allows us to write exact solutions for the ground state entropy $\frac{S}{N} = (\frac{1}{4}+\frac{1}{8L^2})\ln(2)$ and the low temperature Curie tail of the susceptibility, results which are supported numerically.

For ferromagnetic edge-sharing couplings ($J_2<0$), we have shown that at sufficiently low temperatures all combinations of ferromagnetically aligned spins across edge-sharing ($J_2$) bonds are equally realized, resulting in diffuse magnetic scattering about $\vec{q} = (\frac{4\pi}{a},0)$ and $C_6$ rotations, similar to the pure sorrel net case.  In contrast to this pure case, we have shown that the low temperature susceptibility shows a considerably stronger Curie tail, which we have found to be consistent with exact $L=2$ results.  Knowing the ground state, we have found an exact expression for the residual entropy  $\frac{S}{N} = \frac{3\ln(2)}{8}$ of the spin system which is supported numerically.

\section{Acknowledgments}

This work was supported by NSERC (JMH), NSERC USRA (JJB), the Brandon University Research Committee (JJB) and Manitoba Career Focus (JJB).  JMH would like to thank Travis Redpath, John Chalker, Roderich Moessner, Chris Henley and Onofre Rojas for useful comments during the completion of this manuscript.

redpajama_set_name: RedPajamaArXiv


text:
//-----------------------------------------------------------------------
// <copyright file="IncrementalSugiyamaTests.cs" company="Microsoft">
//   (c) Microsoft Corporation.  All rights reserved.
// </copyright>
//-----------------------------------------------------------------------
using System.Collections.Generic;
using System.IO;
using System.Linq;
using Microsoft.Msagl.Core.Layout;
using Microsoft.Msagl.Layout.Layered;
using Microsoft.VisualStudio.TestTools.UnitTesting;

namespace Microsoft.Msagl.UnitTests
{
    /// <summary>
    /// Verifies that incremental sugiyama layout works correctly.
    /// </summary>
    [TestClass]
    [DeploymentItem(@"Resources\DotFiles\LevFiles\chat.dot", "Dots")]
    public class IncrementalSugiyamaTests : MsaglTestBase
    {
        [TestMethod]
        [Description("Verifies that updating shapes and calling incremental sugiyama does not affect the ordering of nodes and layers.")]
        public void NodeShapeChange()
        {
            // Setup
            string filePath = Path.Combine(this.TestContext.TestDir, "Out\\Dots", "chat.dot");
            GeometryGraph graph = this.LoadGraph(filePath);
            var settings = new SugiyamaLayoutSettings();

            // Initial layout
            LayeredLayout layeredLayout = new LayeredLayout(graph, settings);
            layeredLayout.Run();
            SortedList<double, SortedList<double, Node>> originalLayers = SugiyamaValidation.GetLayers(graph, true);

            // Incremental layout
            List<Node> nodes = graph.Nodes.ToList();
            for (int i = 0; i < nodes.Count; i++)
            {
                // Resize a node
                Node node = nodes[i];
                node.BoundaryCurve = node.BoundaryCurve.ScaleFromOrigin(2.0, 2.0);

                // Run incremental layout
                LayeredLayout.IncrementalLayout(graph, node);

                // Verify - the layering and ordering of nodes should not have changed.
                SortedList<double, SortedList<double, Node>> newLayers = SugiyamaValidation.GetLayers(graph, true);
                VerifyLayersAreEqual(originalLayers, newLayers);
            }
        }

        /// <summary>
        /// Verifies that to sets of layers are ordered the same and each layer contains the same order of the same nodes.
        /// </summary>
        private static void VerifyLayersAreEqual(SortedList<double, SortedList<double, Node>> layers1, SortedList<double, SortedList<double, Node>> layers2)
        {
            Assert.AreEqual(layers1.Count, layers2.Count, "The two layer collections have different number of layers.");

            for (int i = 0; i < layers1.Count; i++)
            {
                IList<Node> nodes1 = layers1.Values[i].Values;
                IList<Node> nodes2 = layers2.Values[i].Values;

                Assert.AreEqual(nodes1.Count, nodes2.Count, "The two layers have different number of nodes.");

                for (int j = 0; j < nodes1.Count; j++)
                {
                    Assert.AreEqual(nodes1[j], nodes2[j], "The two layers have different ordered nodes.");
                }
            }
        }
    }
}

redpajama_set_name: RedPajamaGithub


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